Chinese Citizens living in a third country can visit the ROC.
All PRC PASSPORT HOLDERS need to call (415) 362-7680 X133 For an appointment. Personal Appearance is mandatory and Visas are issued on case by case basis. .
If you are in another area of the US, call the nearest Taiwan Agency for details.

The USA green card is an internal American document, giving the holder the right to live in the US. It has no relevance outside the country whatsoever. Your friend will need to do whatever any other PRC national has to do to visit Taiwan.